LETTER
doi:10.1038/nature16944
Controlling spin relaxation with a cavity
A. Bienfait1, J. J. Pla2†, Y. Kubo1†, X. Zhou1,3, M. Stern1,4, C. C. Lo2, C. D. Weis5, T. Schenkel5, D. Vion1, D. Esteve1,
J. J. L. Morton2 & P. Bertet1
versatile control over spin relaxation1. Consider a spin embedded in
a microwave cavity of quality factor Q and frequency ω0. If the cavity
damping rate κ = ω0/Q is greater than the spin–cavity coupling g, then
the cavity provides an additional channel for spontaneous emission of
microwave photons, governed by the Purcell rate6,13
ΓP = κ
g2
κ 2 /4 + δ 2
(1)
in which δ = ω0 − ωs is the spin–cavity detuning (see Fig. 1a and
Methods). This cavity-enhanced spontaneous emission can be much
larger than in free space, and is strongest when the spins and cavity are
a
Cavity-enhanced
radiation
Free-space
radiation
Phonon emission
5 μm
Z0
0
72
C/2
L
C/2
c
z
μm
b
T
x
–0.3
–4
JPA
y
Z0, Pin
B0
π
π/2
0.3
B1
Al wire
z (μm)
Spontaneous emission of radiation is one of the fundamental
mechanisms by which an excited quantum system returns to
equilibrium. For spins, however, spontaneous emission is generally
negligible compared to other non-radiative relaxation processes
because of the weak coupling between the magnetic dipole and
the electromagnetic field. In 1946, Purcell realized1 that the rate
of spontaneous emission can be greatly enhanced by placing the
quantum system in a resonant cavity. This effect has since been used
extensively to control the lifetime of atoms and semiconducting
heterostructures coupled to microwave2 or optical3,4 cavities, and
is essential for the realization of high-efficiency single-photon
sources5. Here we report the application of this idea to spins in solids.
By coupling donor spins in silicon to a superconducting microwave
cavity with a high quality factor and a small mode volume, we reach
the regime in which spontaneous emission constitutes the dominant
mechanism of spin relaxation. The relaxation rate is increased
by three orders of magnitude as the spins are tuned to the cavity
resonance, demonstrating that energy relaxation can be controlled
on demand. Our results provide a general way to initialize spin
systems into their ground state and therefore have applications in
magnetic resonance and quantum information processing6. They
also demonstrate that the coupling between the magnetic dipole
of a spin and the electromagnetic field can be enhanced up to the
point at which quantum fluctuations have a marked effect on the
spin dynamics; as such, they represent an important step towards
the coherent magnetic coupling of individual spins to microwave
photons.
Spin relaxation is the process by which a spin reaches thermal equilibrium by exchanging an energy quantum ħωs with its environment
(where ħ is the reduced Planck constant and ωs is the resonance frequency of the spin), for example in the form of a photon or a phonon,
as shown in Fig. 1a. Understanding and controlling spin relaxation is
essential in applications such as spintronics7, quantum information
processing8, and magnetic resonance spectroscopy and imaging9. For
such applications, the spin relaxation time T1 must be sufficiently
long to permit coherent spin manipulation; however, if T1 is too
long, it becomes a major bottleneck that limits the repetition rate of
an experiment, which in turn affects factors such as the achievable
sensitivity. Certain types of spins can be actively reset to their ground
state by optical10 or electrical11 means, owing to their specific energylevel scheme, and methods such as chemical doping have been used
to influence spin relaxation times ex situ12. Nevertheless, an efficient,
general and tunable initialization method for spin systems is still
currently lacking.
At first inspection, spontaneous emission would appear unlikely to
influence spin relaxation: for example, an electron spin in free space and
at a typical frequency of ωs/(2π) ≈ 8 GHz spontaneously emits photons
at a rate of about 10−12 s−1. However, the Purcell effect provides a way
to markedly enhance spontaneous emission and thus gain precise and
–2
0
y (μm)
2
0
10
Bi concentration
(1016 cm–3)
N1
Echo
N2
Copper box
20 mK
Figure 1 | Purcell-enhanced spin relaxation and experimental set-up.
a, By placing a spin in a resonant cavity, radiative spin relaxation can be
made to dominate over intrinsic processes such as phonon-induced
relaxation. b, Top, a planar superconducting resonator with frequency
ω 0 = 1 / LC consisting of an interdigitated capacitor (black; with a
capacitance C) in parallel with an inductive wire (green; with an
inductance L) is fabricated on top of Bi-doped 28Si. A static magnetic field
B0 is applied parallel to the x–y plane of the 50-nm-thick aluminium layer,
with a tunable orientation θ. Bottom, magnetic field lines of the microwave
excitation field B1 generated by the aluminium wire (arrows) are
superimposed over the local concentration of Bi donors (red), obtained by
secondary ion mass spectrometry (SIMS). c, The sample is mounted in a
copper box that is thermally anchored at 20 mK, and probed by microwave
pulses via asymmetric antennae that are coupled with rate κ1 ≈ κ2/5 to the
resonator. Microwave pulses at ω0 of power Pin are sent by antenna 1, and
the microwave signal leaving via antenna 2 is directed to the input of a
Josephson parametric amplifier (JPA).
1
Quantronics Group, SPEC, CEA, CNRS, Université Paris-Saclay, CEA-Saclay, 91191 Gif-sur-Yvette, France. 2London Centre for Nanotechnology, University College London, London WC1H 0AH, UK.
Institute of Electronics Microelectronics and Nanotechnology, CNRS UMR 8520, ISEN Department, Avenue Poincaré, CS 60069, 59652 Villeneuve d’Ascq Cedex, France. 4Quantum Nanoelectronics
Laboratory, BINA, Bar Ilan University, Ramat Gan, Israel. 5Accelerator Technology and Applied Physics Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA.
†
Present addresses: School of Electrical Engineering & Telecommunications, University of New South Wales, Sydney, New South Wales 2052, Australia (J.J.P.); Okinawa Institute of Science and
Technology (OIST) Graduate University, Onna, Okinawa 904-0495, Japan (Y.K.).
3
7 4 | NAT U R E | VO L 5 3 1 | 3 M A RC H 2 0 1 6
© 2016 Macmillan Publishers Limited. All rights reserved
LETTER RESEARCH
a
7.4
7.3
7.2
Resonator B
Resonator A
1
Transmission, |S21|
b
0.5
68kHz
0
–0.1
2
Echo amplitude (a.u.)
Frequency (GHz)
7.5
23kHz
0.1
(Z – Z0)/(2π) (MHz)
c
1
0
3
4
5
Magnetic field, B0 (mT)
Inversion
W
π
Echo
6
Read-out
AQ(T)
T
π/2
π
2
d
Pin
W
π/2
π
Echo
π
1
)
1.0
AQ(T)/AQ(
Echo amplitude (a.u.)
on resonance (δ = 0): ΓP = 4g2/κ. Furthermore, the Purcell rate can be
modulated by changing the coupling constant or the detuning, allowing
spin relaxation to be tuned on demand.
Although the Purcell effect was used to detect spontaneous emission
of radiofrequency radiation from nuclear spins coupled to a resonant
circuit14, the corresponding Purcell rate ΓP ≈ 10−16 s−1 (or 1 photon
emitted every 300 million years) was negligible compared to the intrinsic spin–lattice relaxation processes. For photon emission to become
the dominant spin-relaxation mechanism, both a large spin–cavity coupling and a low cavity damping rate are needed; in our experiment, this
is achieved by combining the microwave confinement provided by a
micrometre-scale resonator with the high quality factors achieved by
using superconducting circuits.
The device consists of two planar aluminium lumped-element superconducting resonators (denoted A and B) patterned onto a silicon chip
that was purified in nuclear-spin-free 28Si and implanted with bismuth
atoms (see Fig. 1b) at a sufficiently low concentration for collective radiation effects to be absent. A static magnetic field B0 is applied in the
plane of the aluminium resonators, at an angle θ from the resonator
inductive wire, tunable in situ. The device is mounted inside a copper
box and cooled to 20 mK. Each resonator can be used to perform
inductive detection of the electron-spin resonance (ESR) signal of the
bismuth donors: microwave pulses at ω0 are applied at the resonator
input, generating an oscillating magnetic field B1 around the inductive
wire that drives the surrounding spins; the quantum fluctuations of this
field, present even when no microwave is applied, are responsible for
the Purcell spontaneous emission. Hahn echo pulse sequences15 are
used, resulting in the emission of a spin-echo in the detection waveguide, which is amplified with a sensitivity reaching the quantum limit
by a Josephson parametric amplifier16 before demodulation at roomtemperature, yielding the integrated echo signal quadrature AQ (see
Methods). A more detailed description of the set-up is found in ref. 17.
Bismuth is a donor in silicon18 with a nuclear spin I = 9/2. At cryogenic temperatures it can bind an electron (with spin S = 1/2) in addition to those shared with the surrounding Si lattice. The large hyperfine
interaction AS · I between the electron and nuclear spin (in which S
and I are the electron and nuclear spin operators, and A/h = 1.475 GHz
with h the Planck constant) produces a splitting of 7.375 GHz between
the ground and excited multiplets at zero magnetic field (see Fig. 2a
for the complete energy diagram19). This splitting makes the system
ideal for coupling to superconducting circuits20,21. At low fields
(B0 < 10 mT, compatible with the critical field of aluminium), all
ΔmF = ±1 transitions are allowed, where mF is the projection of the
total spin (F = I + S) along B0. Considering only the transitions with
largest matrix element, resonator A (ω 0,A /(2π) = 7.245 GHz,
QA = 3.2 × 105) crosses the |F , mF ⟩ = |4, − 4⟩↔|5, − 5⟩ transition,
whereas resonator B (ω0,B/(2π) = 7.305 GHz, QB = 1.1 × 105) crosses the
transitions |4, − 4⟩↔|5, − 5⟩, |4, − 3⟩↔|5, − 4⟩ and |4, − 2⟩↔|5, − 3⟩
(see Fig. 2a, b).
The echo signal AQ from each resonator as a function of B0 shows
resonances at the expected magnetic fields, split into two peaks each
with a full-width at half-maximum of Δω/(2π) ≈ 2 MHz (see
Fig. 2a). As is explained in ref. 17, this splitting is believed to be the
result of strain induced in the silicon at the donor implant depth of
approximately 100 nm by the aluminium circuit deposited on the surface. In the following, we focus on the lower-frequency peak of the
|4, − 4⟩↔|5, − 5⟩ line, which corresponds to spins lying under the wire.
Over the region occupied by these spins, the amplitude of the B1 field
varies by less than ±2%, as evidenced by the well-defined Rabi oscillations observed when we sweep the power of the refocusing pulse Pin at
the cavity input (see Fig. 2c), which allows us to determine the input
power of a π pulse for a given pulse duration.
We measure the relaxation time T1 by performing an “inversionrecovery” experiment22 (see schematic in Fig. 2d), with the static field
B0 aligned along x (θ = 0). A π pulse first inverts the spins whose
­frequencies lie within the resonator bandwidth κA/(2π) = 23 kHz or
0.5
0
T1 = 1.0 ± 0.2 s
T1 = 0.35 ± 0.1 s
–1
0
0
2
4
6
1/2
P1/2
in (pW )
0
1
2
3
Delay, T (s)
Figure 2 | ESR spectroscopy and Purcell-limited T1 measurement.
a, Top, dominant electron spin resonance transitions of the Si:209Bi spin
system (see Methods). We use two resonators, A (green) and B (brown),
with frequencies of 7.246 GHz and 7.305 GHz, respectively, that cross
up to three spin transitions in the magnetic field range 0–6 mT, as seen
in the echo-detected magnetic field sweep (bottom; vertically offset
for clarity). Subsequent spin relaxation measurements were made
at the magnetic fields indicated by the arrows, corresponding to the
|F , mF ⟩ =|4, − 4⟩↔|5, − 5⟩ transition for each resonator. The doublet
structure of each transition is caused by strain exerted by the aluminium
film on the donors17. b, Cavity linewidths for resonators A and B are found
to be 23 kHz and 68 kHz, respectively, from fits (solid lines) to their
measured transmission amplitude. c, Rabi oscillations are driven by
varying the cavity input power of the refocusing π pulse (5 μs long) applied
τ = 300 μs after the first π/2 pulse. Solid lines are exponentially damped
sinusoidal fits. d, The inversion-recovery sequence is used to measure the
spin relaxation time T1. Spin polarization is measured with a Hahn echo
sequence. AQ is rescaled by its value for T T1 (‘AQ(∞)’) such that it varies
from −1 when the spins are fully inverted to +1 at thermal equilibrium
(see Methods for full sequence description). Data were obtained with the
static field B0 parallel to the inductor (θ = 0). Solid lines are exponential
fits to the data with time constant T1. The uncertainty is provided by the
standard deviation in the exponential fit parameters. a.u., arbitrary units.
In all panels, the symbols represent data for each resonator (A, green
squares; B, brown circles).
κB/(2π) = 68 kHz; this constitutes a small subset of the total number
of spins because κA,B Δω . After a varying delay T, a Hahn echo
sequence provides a measure of the longitudinal spin polarization.
By fitting the data with decaying exponentials, we extract T1 = 0.35 s
for resonator A and T1 = 1.0 s for resonator B.
To quantitatively compare our results with the expected Purcell rate,
it is necessary to evaluate the spin–resonator coupling constant
3 M A RC H 2 0 1 6 | VO L 5 3 1 | NAT U R E | 7 5
© 2016 Macmillan Publishers Limited. All rights reserved
RESEARCH LETTER
g (θ ) = γe⟨F , mF |Sx|F + 1, mF − 1⟩ δB1,2 y cos 2(θ ) + δB1,2 z
(2)
(since δB1,x = 0). This orientation dependence is verified experimentally
by measuring the Rabi frequency as a function of θ, as shown in
Fig. 4a, b, which allows us to extract g(0)/(2π) = 58 Hz and
g(π/2)/(2π) = 17 Hz. As expected, we measure longer spin relaxation
times for increasing values of θ, as shown in Fig. 4c, with the relaxation
1
2
rate T −
1 scaling as [g(θ)] , in agreement with equation (1). Overall, the
data in Figs 3 and 4 demonstrate unambiguously that cavity-enhanced
spontaneous emission is by far the dominant spin-relaxation channel
when the spins are resonant with the cavity, because the probability of
a spin-flip occurring as a result of emission of a microwave photon in
the cavity is 1/[1 + ΓNR/ΓP(δ = 0)] = 0.999, very close to unity.
The spontaneous emission evidenced here is an energy-relaxation
mechanism that does not require the presence of a macroscopic magnetization to be effective. Under the Purcell effect, each spin independently relaxes towards thermal equilibrium by microwave photon
emission, so that when no intra-cavity thermal field is present, the
sample ends up in a fully polarized state after a time longer than Γ −P 1,
regardless of its initial state. This is in stark contrast to the well-known
a
Magnetic-field pulse, BG
Saturation
Read-out
AQ(T)
π/2 π
T
[AQ( if) – AQ(T)] /[AQ( if) – AQ(0)]
b
Echo
100
G/(2π) = 3.8 MHz
10–1
1 MHz
0.38 MHz
0 MHz
10–2
0
0.5
1.0
1.5
2.0
Delay, T (103 s)
c
103
102
T1 (s)
g = γe⟨F , mF |Sx|F + 1, mF − 1⟩ δB⊥ , in which γe/(2π) ≈ 28 GHz T−1 is
the electronic gyromagnetic ratio, Sx is the dimensionless Pauli operator
for the electron spin and δB⊥ is the component of the resonatorfield vacuum fluctuations orthogonal to B 0 (see Methods).
A numerical estimate yields g0/(2π) = 56 ± 1 Hz for the spins located
below the inductive wire in the resonator that are probed in our measurements, and for θ = 0. An independent estimate is obtained by measuring Rabi oscillations: their frequency ΩR = 2g 0 n directly yields g0
given knowledge of the average intra-cavity photon number n , which
can be determined with about 30% imprecision from Pin and the measured resonator coupling to the input and output antennae (see
Methods). Using this method, we obtain g0/(2π) = 50 ± 7 Hz for resonator A and 58 ± 7 Hz for resonator B, compatible with the numerical
estimate. The corresponding Purcell time of the resonant spontaneous
emission is Γ −P 1 = 0.36 ± 0.09 s for resonator A and Γ −P 1 = 0.81 ± 0.17 s
for resonator B, in agreement with the experimental values.
According to equation (1), a Purcell-limited T1 should be strongly
dependent on the spin–cavity detuning. We introduce a pulse in the
magnetic field of duration T between the spin excitation and the spinecho sequence (see Fig. 3a), which results in a temporary detuning δ of
the spins. The amplitude of the echo signal AQ as a function of T yields
their energy relaxation time while they are detuned by δ. To minimize
the influence of spin diffusion22, the spin excitation is performed by a
high-power long-duration saturating pulse (see Fig. 3a and Methods)
instead of an inversion pulse as in Fig. 2d. As is evident in Fig. 3b, we
find that the decay of the echo signal is well fitted by a single exponential with a decay time that increases with |δ|. The extracted T1(δ) curve
(see Fig. 3c) shows an increase in T1 of up to three orders of magnitude
when the spins are detuned away from resonance, until it becomes
limited by a non-radiative energy decay mechanism with characteristic
time Γ −NR1 = 1, 600 ± 300 s. Given the doping concentration in our sample, this non-radiative decay time is consistent with earlier measurements of donor spin relaxation times23, which have been attributed to
charge hopping, but it could also arise here from spatial diffusion of the
spin magnetization away from the resonator mode volume. It is shown
in Fig. 3c that the T1(δ) measurements are in agreement with the
expected dependence (ΓP(δ) + ΓNR)−1, with ΓNR the only free parameter in this fit.
Having demonstrated the effect of cavity linewidth and detuning on
the Purcell rate, we explore the effect of modulating the spin–cavity
coupling constant g. This can be achieved by varying the orientation θ
of the static magnetic field B0 in the x–y plane (Fig. 1b), which adjusts
the component of the microwave magnetic field (B1, which is mostly
aligned along y under the inductive wire) that is orthogonal to B0. More
precisely
101
100
–4
–3
–2
–1
0
1
2
3
4
G/(2π) (MHz)
Figure 3 | Controlling Purcell relaxation by spin–cavity detuning.
a, In between their saturation and subsequent read-out, the spins are
detuned from the cavity by δ = ddωBs Bδ by applying a magnetic-field pulse
with an amplitude of Bδ, with 21π ddωBs ≈ 25 GHz T−1 for this transition and
magnetic field. b, Measured spin-polarization decays (symbols) for four
different detunings δ, which are well fitted by exponential decays (lines),
with relaxation time constants T1 increasing with the detuning (error bars
indicate the standard deviation of a measured echo). c, Measured T1 as a
function of detuning δ (blue symbols). The red line is a fit with
(ΓP(δ) + ΓNR)−1, yielding Γ −NR1 = 1,600 s. Error bars are estimates of the
standard deviation of the fit. These measurements are taken using
resonator B and with θ = π/4, which results in T1 = 1.7 s at δ = 0.
phenomenon of radiative damping24 of a transverse magnetization
generated by earlier microwave pulses, which is a coherent collective
effect under which the degree of polarization of a sample cannot
increase. Had our device possessed a larger spin concentration, spontaneous relaxation would have occurred collectively, manifesting itself
as a non-exponential decay of the echo signal on a timescale faster than
1
6,25
. The existΓ−
P (ref. 13), and leading to an incomplete thermalization
26
ence of such super-radiant or maser emission requires the dimensionless ‘co-operativity’ parameter C = Ng2/(κΔω) (where N is the total
number of spins) to satisfy C 1 (refs 6, 25, 27), which is not the case
here because of the large inhomogeneous broadening of the spin resonance caused by strain.
Our demonstrated ability to modulate spin relaxation through three
orders of magnitude by changing the applied field by less than 0.1 mT
opens up new perspectives for spin-based quantum information processing: long intrinsic relaxation times, which are desirable to maximize
the spin coherence time, can be combined with fast, on-demand initialization of the spin state. Similarly, performing electron spin resonance
at dilution refrigerator temperatures can be prohibitively slow without
the ability to accelerate spin relaxation on demand. We also anticipate
that Purcell relaxation will offer a powerful approach to dynamical
nuclear polarization28,29, for example, by tuning the cavity to match
an electron-nuclear spin flip-flop transition, enhancing the rate of
7 6 | NAT U R E | VO L 5 3 1 | 3 M A RC H 2 0 1 6
© 2016 Macmillan Publishers Limited. All rights reserved
a
0
10
1
b
60
g/(2π) (Hz)
LETTER RESEARCH
40
1/2
Pin
(pW1/2)
8
6
4
20
2
0
0
40
80
0
0.5
1
1.0
0°
30°
40°
50°
60°
0
–1
2
1/T1 (s–1)
AQ(T)/AQ(
)
c
1
cos(T)
T (°)
0.5
0
0
4
6
0.5
cos2(T)
8
1
10
Delay, T (s)
Figure 4 | Dependence of Purcell relaxation on spin–cavity coupling g.
a, Rabi oscillations (as in Fig. 2c) measured as a function of field
orientation θ (see Fig. 1b); the colour scale indicates the echo amplitude in
arbitrary units. b, The Rabi oscillations in a are used to extract the spincavity coupling strength g (blue symbols; error bars are determined by
the 30% accuracy on Pin). These data are fit to equation (2) (red line); the
non-zero value of g(π/2) is due to the finite out-of-plane component of the
microwave magnetic field. c, Inversion-recovery measurements (error bars
indicate the standard deviation of a measured echo) for different values of
θ confirm that the relaxation time T1 (see inset; error bars are estimates of
the standard deviation of the fit) varies as [g(θ)]2. The red line in the inset
is the Purcell formula predicted using the g(θ) dependence fitted from b.
All data were collected using resonator B.
cross-relaxation to pump polarization into the desired nuclear spin
state30 (see Methods). The Purcell rate we obtain could be increased
by reducing the transverse dimensions of the inductor wire to yield
larger coupling constants (up to 5–10 kHz), which would reduce the
spontaneous emission time to less than 1 ms (thus permitting faster
repetition rates and a higher sensitivity17), allowing for the possibility
of high-co-operativity coupling of a single spin to the microwave cavity
field. Our measurements constitute evidence that vacuum fluctuations
of the microwave field can affect the dynamics of spins, and, therefore, are a step towards the application of concepts in circuit quantum
electrodynamics to individual spins in solids.
Online Content Methods, along with any additional Extended Data display items and
Source Data, are available in the online version of the paper; references unique to
these sections appear only in the online paper.
Received 24 August; accepted 11 December 2015.
Published online 15 February 2016.
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Acknowledgements We acknowledge technical support from P. Sénat, D. Duet,
J.-C. Tack, P. Pari and P. Forget, as well as discussions within the Quantronics
group. We acknowledge support of the European Research Council under the
European Community’s Seventh Framework Programme (FP7/2007-2013)
through grant agreements No. 615767 (CIRQUSS), 279781 (ASCENT) and
630070 (quRAM), and of the C’Nano IdF project QUANTROCRYO. J.J.L.M. is
supported by the Royal Society. C.C.L. is supported by the Royal Commission for
the Exhibition of 1851. T.S. and C.D.W. were supported by the US Department
of Energy under contract DE-AC02-05CH11231.
Author Contributions A.B., J.J.P., J.J.L.M. and P.B. designed the experiment.
X.Z. and D.V. designed and fabricated the Josephson Parametric Amplifier.
C.C.L., C.D.W. and T.S. provided the bismuth-implanted isotopically purified
silicon sample. A.B., J.J.P. and Y.K. fabricated the sample and performed the
measurements. A.B., J.J.P., Y.K., J.J.L.M. and P.B. analysed the data. J.J.L.M., D.E.,
D.V. and P.B. supervised the project. A.B., J.J.P., Y.K., M.S., D.V., D.E., J.J.L.M. and
P.B. contributed to writing the paper.
Author Information Reprints and permissions information is available at
www.nature.com/reprints. The authors declare no competing financial
interests. Readers are welcome to comment on the online version of the
paper. Correspondence and requests for materials should be addressed to
P.B. (patrice.bertet@cea.fr).
3 M A RC H 2 0 1 6 | VO L 5 3 1 | NAT U R E | 7 7
© 2016 Macmillan Publishers Limited. All rights reserved
RESEARCH LETTER
METHODS
Bismuth donors in silicon. Bismuth donors in silicon have the following isotropic
spin Hamiltonian31: Hˆ / h = B ⋅ (γeS ⊗ − γn ⊗ I )+ AS ⋅ I, in which the electronic
gyromagnetic ratio γe/(2π) = 27.997 GHz T−1, the nuclear gyromagnetic ratio
γn/(2π) = 6.9 MHz T−1 and the hyperfine coupling constant A/h = 1.475 GHz. For
a weak static field B0 (B0  50 mT) oriented along x, the eigenstates of the total
angular momentum F = S + I and its projection mF along B0 represent good quantum numbers for the 20 electro-nuclear energy states of the Bi:Si system. These
eigenstates can be grouped in an F = 4 ground and an F = 5 excited multiplet separated by a frequency of (I + 1/2)A/h = 7.35 GHz in zero-field (see Fig. 1d).
Transitions between states that verify ΔFΔmF = ±1 can be excited with a field
orientated along y (or z) because their associated matrix element
⟨F , mF |S y|F + 1, mF ± 1⟩=⟨F , mF |S z|F + 1, mF ± 1⟩ has the same magnitude as an
ideal electronic spin 1/2 transition ⟨msSyms′⟩ = 0.5. Only the ten transitions with
a matrix element greater than 0.25 are shown in Fig. 2a. Characteristics for the
transitions probed by our resonators are given in Extended Data Table 1.
Single-spin coupling to the resonator. The spin–resonator interaction is described
by a Jaynes–Cummings Hamiltonian, ħg(a†σ− + aσ+), in which a (a†) is the field
annihilation (creation) operator, σ− (σ+) is the spin lowering (raising) operator
and g is the spin–resonator coupling strength. For the Bi:Si transitions,
|F , mF ⟩↔|F + 1, mF − 1⟩ probed by the resonators, g can be expressed as
g = γe⟨F , mF |Sx|F + 1, mF − 1⟩ δB⊥ (ref. 17), in which δB⊥ is the component of
the resonator-field vacuum fluctuations orthogonal to B0. Considering the orientations for B0 and δB shown in Fig. 1b, we obtain equation (2)
g(θ )= γe⟨F , mF |Sx|F + 1, mF − 1⟩ δB1,2 y cos(θ )2 +δB1,2 z
To estimate the distribution of the coupling constant g(0) for a given transition,
we need to estimate the vacuum-field fluctuations δB1 in the spin ensemble region.
This is achieved using the COMSOL software and assuming a non-homogeneous
current-density distribution in the superconducting aluminium wire32. The total
current flowing through the wire cross-section is δi = ω 0 /2Z 0 , in which
Z 0 = L /C is the resonator impedance, determined to be 44 Ω via electromagnetic
simulations realized in CST Microwave Studio. In all the work presented in the
main text, the measurements were done on the low-field peak of transition
|F , mF ⟩ =|4, − 4⟩↔|5, − 5⟩, which has been attributed to spins residing under the
wire. From the spin implantation profile (see Fig. 1b) and the spatial dependence
of the microwave field δB1 restrained to the area under the wire (|y| < 2.5 μm), the
relevant coupling-constant distribution can be extracted. Doing so yields a very
asymmetric coupling distribution that is sharply peaked around g/(2π) = 56 Hz
with a 2-Hz full-width at half-maximum for the transition |4, − 4⟩↔|5, − 5⟩ with
θ = 0. A more detailed derivation of the coupling constant and its estimate at θ = 0°
is available in ref. 17.
Average intra-cavity photon number n. The average intra-cavity photon number n
4κ1P in
of a pulse of power Pin at the cavity input is evaluated as n =
,
ω 0(κ1 + κ 2 + κ L)2
in which κ1 and κ2 are the couplings to the input and output antennas and κL
represents the internal losses of the resonator. From a previous calibration of the
experimental setup17, we estimate that we can determine Pin with an accuracy of
approximately 1 dB. The values of κ1, κ2 and κL are determined experimentally by
measuring each element of the resonator scattering matrix and fitting to the wellknown input–output formulae33; see Extended Data Table 2.
Data acquisition and echo signal. The full description of the experimental set-up
is available in ref. 17. The use of a Josephson parametric amplifier allows us to reach
a quantum-limited sensitivity. In addition to the Hahn echo sequence, we use a
Carr–Purcell–Meiboom–Gill sequence34 for every echo acquisition. For all AQ data
points presented in this work, 10 π pulses are added after the first echo to recover
10 extra echoes, which are subsequently averaged to boost the signal-to-noise ratio.
This scheme allows us to acquire data in single-shot read-out. Each AQ data point
is a single-shot measurement and the error bars are determined by the variance of
a pool of at least n = 200 measurements, taken in similar conditions.
Experimental determination of T1 at resonance. The inversion-recovery
sequence is used to measure the spin relaxation time T1; see Fig. 2d. Spin polarization is measured with the following Hahn echo sequence: 50-μs-long π/2 pulse,
delay τ = 500 μs, and 100-μs-long π pulse. The pulse durations were chosen such
that only spins within a narrow spectral range were detected, producing a welldefined Purcell-limited T1. Indeed, because the probed ensemble of spins has a
larger linewidth Δω = 2 MHz than do our resonators, the signal emitted during
the spin-echo comes from a subset of the ensemble of spins, with a frequency
spectrum at least as large as the resonator bandwidth. Spins probed at the edges
of the bandwidth of the resonator will have longer Purcell relaxation times; for
instance, those detuned by δ = κ have an expected Purcell relaxation time that
is five times slower than the T1 time expected at perfect resonance; see Extended
Data Fig. 1a. The contribution of those spins with a longer decay time to the signal
will result in an averaging effect, meaning that the measured T1 will be erroneously
longer than predicted.
To suppress this effect, we reduce the bandwidth of the read-out sequence to
collect signal only from spins very close to the resonance. The response function
of a pulse of length tp incident on a cavity with bandwidth κ at frequency ω0 is
expressed as
R(ω) = [2sinc(tp(ω − ω 0)/2)]2 R cav(ω) =
[2sinc(tp(ω − ω 0)/ 2)]2
1 + 4(ω − ω 0)2/κ 2
in which sinc(x) = sin(x)/x. As shown in Extended Data Fig. 1a, for the narrowest
bandwidth κ/(2π) = 23 kHz of resonator A, pulses of 5 μs are heavily filtered by
the resonator and have the same bandwidth, whereas 100-μs-long pulses have a
reduced bandwidth of approximately 10 kHz. In case of 100-μs-long excitation
pulses, the Rabi frequency is such that only spins with |δ|/(2π) ≤ 5 kHz will contribute to the signal. This corresponds to a dispersion of only 5% for the expected
Purcell relaxation times, which is negligible. To illustrate the averaging effect,
two inversion-recovery curves are shown in Extended Data Fig. 1b, with readout
pulses of 5 μs and 100 μs. The former yields T1 = 0.65 s, which is a factor two higher
than predicted by the Purcell effect, whereas the latter yields the expected value
T1 = 0.35 s.
Therefore, Fig. 2d shows an inversion-recovery sequence that has a read-out
echo sequence with a narrow bandwidth (tπ = 100 μs, tπ/2 = tπ/2) to suppress contributions from spins with a lower decay rate, and an inversion pulse with a large
bandwidth (tπ = 5 μs) to maximize the efficiency of the inversion.
Given that the spin energy relaxation time T1 is of the order of 1 s, we choose
a repetition rate γrep that is sufficiently low to allow full relaxation of the spins
between successive inversion-recovery sequences: γrep = 0.04 Hz.
Experimental determination of spin–cavity-detuning-dependent relaxation
rates. The spins are detuned from the cavity by applying an additional bias pulse
on one of the Helmholtz coils used to apply the static field B0. The extra bias
pulse is output by a pulse generator with 50 Ω output impedance placed in parallel to the d.c. supply of one of the Helmholtz coils. To minimize the effect of
transients due to the 1-Hz bandwidth of the coils, buffer times of 1 s are added
after ramping the coil up and down. To limit the loss of signal during these
buffer times, we use an angle θ = 45° and work with resonator B in order to
have a longer T1(0) = 1.68 s. Applying a magnetic-field pulse to a single coil
instead of both coils perturbs θ by at most 4°. The value of T1(0) was measured
with inversion recovery. All the data presented in Fig. 3 and in Extended Data
Fig. 2 were acquired in a separate run. The quality factor of resonator B decreased
from Q = 1.07 × 105 to Q = 8.9 × 104 owing to slightly higher losses, yielding the
resonator bandwidth κ/(2π) = 82 kHz.
To observe the long relaxation times, such as those measured in Fig. 3, inversion recovery is not an ideal method. When the spin linewidth is broader (about
20 times) than the excitation bandwidth and when the thermalization time is very
long, one can observe polarization mixing mechanisms35,36, spectral and spatial
spin diffusion being the most relevant to our case, because the spin system is composed of only one species. If we try to measure the relaxation from spins that have
been detuned by an amount δ/(2π) = (ωs − ω0)/(2π) = 3.8 MHz during a lapse of
time T with an inversion-recovery sequence (Extended Data Fig. 2a), then we
observe a double-exponential relaxation (Extended Data Fig. 2d, green), which
we attribute to the existence of a spin-diffusion mechanism.
Spin diffusion is prevented by suppressing any polarization gradient along the
spin line, which leads us to use a saturation-recovery scheme instead of inversion recovery. The simplest saturation-recovery scheme (Extended Data Fig. 2b)
consists of sending a strong microwave tone that results in the saturation of the
line, producing an incoherent mixed state with the population evenly distributed
between excited and ground states. Nevertheless, a relaxation time measured using
this scheme still yields a double-exponential decay (Extended Data Fig. 2d, orange),
with time constants similar to those for the inversion recovery case. This implies
that the saturation of the line is insufficient.
To improve the saturation, we can sweep the magnetic field during the saturation pulse to bring different subsets of the spin line to resonance and realize a full
saturation. The adopted sweep scheme is shown in Extended Data Fig. 2c. The
corresponding relaxation curve fits well to a simple exponential decay (Extended
Data Fig. 2d, blue), indicating the suppression of the spin-diffusion effect.
We further check the quality of the saturation by measuring the polarization
across the full spin linewidth immediately after saturation. To realize such scans
(Extended Data Fig. 2e), we apply the relevant saturation pulse at ω0, then apply a
magnetic field pulse Bδ = (ωs − ω0)/γe and measure the echo signal AQ(ωs) with a
Hahn echo sequence. When no saturation pulse is applied, the measured echo
© 2016 Macmillan Publishers Limited. All rights reserved
LETTER RESEARCH
signal AQ0(ωs) is a measure of the full polarization −⟨S z(ωs)⟩=+ 1 (Extended Data
Fig. 2e, black curve) and shows the natural spin linewidth. When studying an
excitation pulse, the polarization of the spins is −⟨S z(ωs)⟩= AQ (ωs)/AQ0 (ωs), in
which AQ(ωs) is the measured echo signal. Therefore, −⟨S z(ωs)⟩=− 1 indicates
full inversion, ⟨S z(ωs)⟩ = 0 indicates saturation and −⟨S z(ωs)⟩=+ 1 indicates
return to thermal equilibrium. The green, orange and blue curves are taken after
a π pulse, after saturation without field sweep and after saturation with field sweep,
respectively. At resonance, we expect a change of Sz from −1 to +1 for a π pulse
and from −1 to 0 for a saturation pulse. Owing to the coil transient time, all three
curves show a partial relaxation. If the saturation was optimal and no partial relaxation was occurring, then we should observe Sz = 0 for any detuning δ. For the two
saturations (with and without field sweep) studied here, only that with field sweep
equally saturates the line. The basic saturation with field sweep has a bandwidth
of approximately 250 kHz and the bandwidth of the π pulse is similar to that of the
cavity κ/(2π) = 82 kHz. This finding confirms that spin diffusion is fully suppressed
only in a scheme of saturation with field sweep, to yield a simple exponential-decay
relaxation. This scheme is used to measure the 22 relaxation rates at different
detunings δ shown in Fig. 4.
The global fit shown in Fig. 4c is obtained using [T1(δ)]−1 = ΓP + ΓNR, which
may be expressed as [T1(0)]−1 [1 + 4(δ/κ )2 ]−1 + ΓNR to involve only experimentally
determined parameters. Indeed, κ is precisely determined by measuring the quality factor of the resonator at low power, and T1(0) is determined by an inversionrecovery sequence, as mentioned above. The parameter δ was determined via precise calibration of the coil pulse. Therefore, the only remaining free parameter in
the fit is ΓNR, yielding Γ −NR1 = 1, 600 s . The error bars come from the accuracy of
the fits of the relaxation rates.
Practical considerations for the application of cavity-induced relaxation in
magnetic resonance. The experiments described in the main text take place at
low magnetic field (B0 < 10 mT) and low temperature (T ≈ 20 mK) using a dilution
refrigerator. These are unusual conditions for magnetic resonance measurements;
however, as we discuss below, with straightforward modifications, cavity-induced
relaxation could be observed in other environments, broadening the class of spin
systems that could be used.
Superconducting microresonators can withstand large magnetic fields (up to
approximately 1 T) while maintaining a large quality factor (Qi ≈ 2 × 105)37–39 if
they are patterned in metals such as Nb, NbN or NbTiN, instead of Al, as we have
used. The use of these alternative metals would enable our results to be applied
to a much larger class of spin systems, including typical electron spins with g ≈ 2.
Similar observations can be made for temperature: Nb, NbN and NbTiN have a
higher critical temperature than does Al, which would permit the use of temperatures of 1–4 K (accessible with conventional liquid helium cryostats). However,
temperature is important for reasons other than helping to maintain a small κ,
because the Purcell effect brings spins into thermal equilibrium with the cavity
field; for example, at the microwave frequencies used in our experiments (7.3 GHz),
temperatures below 70 mK are required for a spin polarization of >99%. Higher
temperatures could be used at the cost of the degree of spin polarization, but this
issue could be addressed by moving to higher frequencies. A third factor when
considering the operating temperature is that cavity-induced relaxation can only
be exploited when it dominates over intrinsic processes such as spin-lattice relaxation. For most spin systems, this requirement translates into temperatures similar
to those of liquid-helium.
The possibility of cavity-induced relaxation with conventional electron spin
systems might lead to applications other than those that benefit from a faster
return to thermal equilibrium to increase signal averaging rates. In particular, we
consider the possibility of cavity-assisted dynamic nuclear polarization, via either
the so-called solid effect or the Overhauser effect, which was recently observed
in solids40. With the solid effect, the equilibrium polarization of a nuclear spin of
frequency ωn coupled to an electron spin of frequency ωe is enhanced by irradiating
it with microwaves at ωe + ωn, provided the electron spins return quickly enough
to equilibrium. Tuning a cavity on resonance with the electron spin transition at
ωe could provide an alternative relaxation mechanism to phonons, thereby avoiding, for example, phonon bottleneck effects and/or mitigating the need to apply
large magnetic fields. With the Overhauser effect, saturating the spin transition
by applying microwaves at ωs enhances the nuclear spin polarization because of
the existence of electron-nuclear spin cross-relaxation processes, which could be
enhanced by tuning a cavity at ωe − ωn.
Sample size. No statistical methods were used to predetermine sample size.
31. Wolfowicz, G. et al. Atomic clock transitions in silicon-based spin qubits. Nature
Nanotechnol. 8, 561–564 (2013).
32. Van Duzer, T. & Turner, C. W. Principles of Superconductive Devices and Circuits
2nd edn (Prentice-Hall PTR, 1999).
33. Palacios-Laloy, A. Superconducting Qubit in a Resonator: Test of the Leggett-Garg
Inequality and Single-shot Readout. PhD thesis, Université Pierre et Marie
Curie — Paris VI (2010).
34. Mentink-Vigier, F. et al. Increasing sensitivity of pulse EPR experiments using
echo train detection schemes. J. Magn. Reson. 236, 117–125 (2013).
35. Bloembergen, N. On the interaction of nuclear spins in a crystalline lattice.
Physica 15, 386–426 (1949).
36. Abragam, A. Principles of Nuclear Magnetism Ch. IX (Oxford Univ. Press, 1983).
37. de Graaf, S. E., Davidovikj, D., Adamyan, A., Kubatkin, S. E. & Danilov, A. V.
Galvanically split superconducting microwave resonators for introducing
internal voltage bias. Appl. Phys. Lett. 104, 052601 (2014).
38. Wisby, I. et al. Coupling of a locally implanted rare-earth ion ensemble
to a superconducting micro-resonator. Appl. Phys. Lett. 105, 102601
(2014).
39. Samkharadze, N. et al. High kinetic inductance superconducting nanowire
resonators for circuit QED in a magnetic field. Preprint at http://arXiv.org/
abs/1511.01760 (2015).
40. Can, T. V. et al. Overhauser effects in insulating solids. J. Chem. Phys. 141,
064202 (2014).
© 2016 Macmillan Publishers Limited. All rights reserved
RESEARCH LETTER
0.6
4
0.4
2
0.2
-40
-20
0
20
40
0
Detuning (kHz)
1.0
)
0.8
0.0
b
8
BW Pulse π 5us
BW resonator
BW Pulse π
100us
6
T1 as given by Purcell law (s)
Normalized response (a.u.)
1.0
AQ(T) /AQ(T=
a
0.5
Pulse π 5us, T1 = 0.65 s
Inversion
0.0
Readout
AQ(T)
T
π /2 π
π
Pulse π 100us, T1=0.35s
-0.5
Inversion
Readout
AQ(T)
T
π
-1.0
0
1
2
3
π /2 π
4
5
6
Time T (s)
Extended Data Figure 1 | Effect of excitation-pulse bandwidth on
the measurement of T1. a, The red and blue lines shown the computed
pulse bandwidth (‘normalized response’) for a 5-μs π pulse and a 100-μs
π pulse, respectively, incident on a cavity with κ/(2π) = 23 kHz (green
dashes). To illustrate the averaging effect of the pulse bandwidth on T1
measurements, the expected Purcell T1 curve (black line) as a function
of spin–cavity detuning is plotted on the right axis, with T1(0) = 0.35 s
and κ/(2π) = 23 kHz. b, T1 measurements for two different π-pulse lengths
(see insets), measured on resonance with resonator A. Spin polarization is
measured with a Hahn echo sequence and AQ is rescaled by its value for
T T1 (‘AQ(T = ∞)’). Symbols are data and solid lines are exponential fits.
The 100-μs π pulse (blue) yields T1 = 0.35 s, which is in agreement with
the Purcell rate. The 5-μs π pulse (red) yields T1 = 0.65 s, a factor of two
greater than the accurate value.
© 2016 Macmillan Publishers Limited. All rights reserved
LETTER RESEARCH
a
c
π, 5µs
Saturation
AQ(T)
AQ(T)
b
Magnetic field pulse Bδ
Readout
Saturation 1s
Pin= - 35.8 dBm
AQ(T)
1s π /2 π
T
1s
echo
e
1
6
5
4
8
3
2
0.1
6
5
0
T1 = 1300s
T1A = 1100s, T1B = 190s
T1A = 800s, T1B = 110s
0.5
0
2
4
6
Time (s)
1.0
AQ(ωs)/AQ0(ω0)
[AQ( if)-AQ(T)] / [AQ( if)-AQ(0)]
d
8
2
1
0
-1
-2
echo
1s π /2 π
T
1s
echo
(ωs-ω0)/2π (MHz)
1s π /2 π
T
1s
Readout
Magnetic field pulse Bδ
Readout
Magnetic field pulse Bδ
0.8
0.6
0.4
0.2
1.5
1
2
Time (103s)
Extended Data Figure 2 | Spectral spin diffusion. a–c, T1 measurement
sequence when spins are detuned from the cavity by applying a magnetic
field Bδ, providing a detuning of δ = ωs − ω0 = 2πγeffBδ, with
γeff = df/dB(B0) the effective gyromagnetic ratio, evaluated as the
derivative of f = 2πωs with respect to the applied magnetic field B at a
given magnetic field B0. In a, a 5-μs π pulse is used to realize an inversionrecovery sequence; in b, a 1-s-long strong microwave pulse sent at cavity
resonance is used to realize a saturation-recovery sequence; in c, a
magnetic field scan (bottom panel) is used in addition to a 6-s-long strong
microwave pulse to realize a saturation-recovery sequence. The expected
magnetic field profile due to the coil filtering, assuming that the coil is an
order-one low-pass filter with a bandwidth of 1 Hz, is shown in orange
(c, bottom panel). d, T1 measurements for sequences shown in a (green),
-1.0
-0.5
0.0
0.5
1.0
(ωs-ω0)/2π (MHz)
b (red) and c (blue) for δ/(2π) = 3.8 MHz. The fits (black lines) to the
green and red data have a double-exponential decay, whereas the fit to the
blue data is a simple exponential. We attribute the double-exponential
decay (with extracted characteristic times T1A and T1B) to spin diffusion.
e, Spectral profiles of the excitation pulse sequences shown in a (green),
b (red) and c (blue). The sequence is as follows: send the excitation pulse,
detune the spins and measure AQ(ωs). The black line is the reference
profile without any excitation pulse, yielding the reference polarization
⟨Sz (ωs)⟩= − AQ0 (ωs)/AQ0 (ωs) =− 1. When an excitation pulse is sent,
we can access ⟨Sz (ωs)⟩=− AQ (ωs)/AQ0 (ωs). To conserve the line shape
profile, we plotted AQ(ωs)/AQ0(ω0) instead of AQ(ωs)/AQ0(ωs). Neither the
π profile nor the saturation profiles reach the full inversion +1 or the full
saturation 0 at resonance; this is an artefact due to the coil transient time.
© 2016 Macmillan Publishers Limited. All rights reserved
RESEARCH LETTER
Extended Data Table 1 | Relevant Bi:Si transitions and their characteristics
© 2016 Macmillan Publishers Limited. All rights reserved
LETTER RESEARCH
Extended Data Table 2 | Resonator characteristics
© 2016 Macmillan Publishers Limited. All rights reserved